### Pareto-optimality

Pareto efficiency, or Pareto optimality, is a state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off. The term is named after Vilfredo Pareto (1848–1923), an Italian economist who used the concept in his studies of economic efficiency and income distribution. The concept has applications in academic fields such as economics and engineering.

Given an initial allocation of goods among a set of individuals, a change to a different allocation that makes at least one individual better off without making any other individual worse off is called a Pareto improvement. An allocation is defined as "Pareto efficient" or "Pareto optimal" when no further Pareto improvements can be made.

For example, suppose there are two consumers A & B and only one resource X. Suppose X is equal to 20. Let us assume that the resource has to be distributed equally between A and B and thus can be distributed in the following way: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7), (8,8), (9,9), (10,10). At point (10,10) all resources have been exhausted. No further distribution is possible - if redistribution continues, it will lead to a position (11,9) or (9,11) that makes one better off and the other worse off. Hence, point (10,10) is Pareto optimal; no further Pareto improvements can be made.

Pareto efficiency is a minimal notion of efficiency and does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society.[1][2] The notion of Pareto efficiency can also be applied to the selection of alternatives in engineering and similar fields. Each option is first assessed under multiple criteria and then a subset of options is identified with the property that no other option can categorically outperform any of its members.

## Pareto efficiency in short

It is commonly accepted[by whom?] that outcomes that are not Pareto efficient are to be avoided, and therefore Pareto efficiency is an important criterion for evaluating economic systems and public policies. If economic allocation in any system is not Pareto efficient, there is potential for a Pareto improvement—an increase in Pareto efficiency: through reallocation, improvements can be made to at least one participant's well-being without reducing any other participant's well-being.

It is important to note, however, that a change from an inefficient allocation to an efficient one is not necessarily a Pareto improvement. Thus, in practice, ensuring that nobody is disadvantaged by a change aimed at achieving Pareto efficiency may require compensation of one or more parties. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive and more efficient, the monopolist will be made worse off. However, the loss to the monopolist will be more than offset by the gain in efficiency. This means the monopolist can be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement.

In real-world practice, such compensations have unintended consequences. They can lead to incentive distortions over time as agents anticipate such compensations and change their actions accordingly. Under certain idealized conditions, it can be shown that a system of free markets will lead to a Pareto efficient outcome. This is called the first welfare theorem. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu. However, the result only holds under the restrictive assumptions necessary for the proof (markets exist for all possible goods so there are no externalities, all markets are in full equilibrium, markets are perfectly competitive, transaction costs are negligible, and market participants have perfect information). In the absence of perfect information or complete markets, outcomes will generically be Pareto inefficient, per the Greenwald–Stiglitz theorem.[3]

## Weak Pareto efficiency

A "weak Pareto optimum" (WPO) is an allocation for which there are no possible alternative allocations whose realization would cause every individual to gain. Thus an alternative allocation is considered to be a Pareto improvement only if the alternative allocation is strictly preferred by all individuals. When contrasted with weak Pareto efficiency, a standard Pareto optimum as described above may be referred to as a "strong Pareto optimum" (SPO).

Weak Pareto-optimality is "weaker" than strong Pareto-optimality in the sense that any SPO also qualifies as a WPO, but a WPO allocation is not necessarily an SPO.

## Use in engineering

The notion of Pareto efficiency is also useful in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

The Pareto frontier is defined formally as follows. Consider a design space with n real parameters (corresponding to the allocation of goods in the economics interpretation), and for each design space point there are m different criteria by which to judge that point (corresponding to the utility of the different agents in the economics interpretation). Let be the function which assigns, to each design space point x, a criteria space point f(x). This represents the way of valuing the designs. Now, it may be that some designs are infeasible; so let X be a set of feasible designs in $\left\{\mathbb\left\{R\right\}\right\}^n$, which must be a compact set. Then the set which represents the feasible criterion points is f(X), the image of the set X under the action of f. Call this image Y. Now construct the Pareto frontier as a subset of Y of the feasible criterion points. It is often assumed in engineering that the preferable values of each criterion parameter are the lesser ones (e.g. lower emissions or lower cost), thus minimizing each dimension of the criterion vector is desired. Then compare criterion vectors as follows: One criterion vector y strictly dominates (or "is preferred to") a vector y* if each parameter of y is not strictly greater than the corresponding parameter of y* and at least one parameter is strictly less: that is, $\mathbf\left\{y\right\}_i \le \mathbf\left\{y*\right\}_i$ for each i and $\mathbf\left\{y\right\}_i < \mathbf\left\{y*\right\}_i$ for some i. This is written as $\mathbf\left\{y\right\} \prec \mathbf\left\{y*\right\}$ to mean that y strictly dominates y*. Then the Pareto frontier is the set of points from Y that are not strictly dominated by any other point in Y. Formally, this defines a partial order on Y, namely the product order on $\mathbb\left\{R\right\}^m$ (more precisely, the induced order on Y as a subset of $\mathbb\left\{R\right\}^m$), and the Pareto frontier is the set of maximal elements with respect to this order.

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science, sometimes referred to as the maximum vector problem or the skyline query.[4][5]